Notes, summaries, assignments, exams, and problems for Mathematics

Understanding Quartiles

Quartiles divide a set of ordered data into four groups with equal numbers of values. The three dividing points are Q1, the median (Q2), and Q3.

• Interquartile Range (IQR): Defined as Q3 – Q1, this represents the range of the middle half of the data. It provides a measure of spread by showing how closely the data are clustered around the median.
• Semi-interquartile Range: One half of the interquartile range.

Quartile Formulas

• Q2 (Median): (n+1) / 2
• Q1: (n+1) / 4
• Q3: 3(n+1) / 4

Outlier Formula

To identify outliers, use the following boundaries:

• Left Boundary: Q1 - 1.5(IQR)
• Right Boundary: Q3 + 1.5(IQR)

Deviation and Standard Deviation

Deviation tells you how far a single data value is from the mean (the difference between a value... Continue reading "Understanding Quartiles, Standard Deviation, and Percentiles" »

1. Type I and Type II Errors

Type I Error (α): Occurs when a researcher rejects a true null hypothesis (a "false positive").

Type II Error (β): Occurs when a researcher fails to reject a false null hypothesis (a "false negative").

The goal of statistical testing is to minimize both errors simultaneously.

2. Parametric vs. Non-Parametric Statistics

Parametric Tests: These assume data is normally distributed and use interval/ratio scales (e.g., t-test, ANOVA).

Non-Parametric Tests: These are "distribution-free" tests used for nominal/ordinal data or small samples (e.g., Chi-square, Mann-Whitney U).

Parametric tests are generally more powerful if their assumptions are met.

3. Null Hypothesis (H₀) vs. Alternative Hypothesis (H₁)

Null Hypothesis (H₀)

Chapter 1: Understanding Variables

Types of Variables

• Categorical: Smoker (current, former, no)
• Ordinal: Non, light, moderate, heavy smoker (ordered categories)
• Quantitative: BMI, Age, Weight (numerical measurements)

Key Definitions

• Observation: Measurements are made (individual or aggregate).
• Variable: The generic characteristic we measure (e.g., age).
• Value: A realized measurement (e.g., 27).

Chapter 2: Statistical Studies

Surveys: Census and Sampling

• Goal: Describe population characteristics.
• Census: Attempts to reach the entire population (costly, time-consuming).
• Sampling: Uses a sample of the population (allows for inferences, saves time and money).
• Simple Random Sampling: Based on probability.
• Issues with Sampling: Under-coverage, volunteer bias,

Investment Portfolio and CAPM Practice Solutions

• Question 1: Portfolio beta (0.6)
• Question 2: Expected return under CAPM (7%)
• Question 3: Expected return with new allocation (6.5%)
• Question 4: Passive investment strategy (Zero alpha)
• Question 5: Net Asset Value calculation ($50)
• Question 6: Definition of NAV (Market value of assets minus liabilities divided by shares outstanding)
• Question 7: Portion of market risk (64%)
• Question 8: Mutual fund return after expenses (4.5%)
• Question 9: Net Asset Value City Street Fund ($40)
• Question 10: Impact on portfolio value (Decreases to $460 million)
• Question 11: Impact on shares outstanding (Decreases to 9 million)
• Question 12: New NAV after shares sold ($40)
• Question 13: Risk premium for stocks (60%)
• Question 14: Optimal

Common Statistical Biases

• Sampling bias: The sample was not representative of the population.
• Non-response bias: Only 24% returned surveys.

Sampling Techniques

• Simple Random Sampling (SRS): 1) Every member of the population has the same chance of being included (representative). 2) Members are chosen independently.
• Random Cluster Sampling: 1) Divide into smaller geographical sectors. 2) Take an SRS of sectors. 3) Count all samples in sectors and scale appropriately.
• Stratified Random Sampling: 1) Divide population into groups based on criteria like age or income. 2) Perform an SRS of each group and scale appropriately.

Data Variables and Distributions

• Variable Types: Categorical and Numeric (discrete and continuous).
• Relative frequency: Count / sample

Set Difference Calculation

To find the set difference A - B, we identify all elements present in set A but not in set B.

Step-by-Step Subtraction

• Is 1 ∈ B? No. (Keep 1)
• Is 2 ∈ B? No. (Keep 2)
• Is 3 ∈ B? Yes. (Remove 3)
• Is 5 ∈ B? No. (Keep 5)
• Is 7 ∈ B? Yes. (Remove 7)
• Is 8 ∈ B? No. (Keep 8)

The remaining elements from set A are {1, 2, 5, 8}.

Symbolic Logic

In symbolic logic, the word "but" functions like "and," indicating that both conditions occur simultaneously. To write "He is rich but not generous" in symbolic form:

• p: "He is rich"
• q: "He is generous"
• ¬q: "He is not generous"
• ∧: The conjunction operator

Logic Symbol Reference

Logarithm Calculations

To find the value of log 360,... Continue reading "Step-by-Step Solutions for Mathematical Problems" »

Parametric Inference Fundamentals

The probability distribution of the population under study is known, except for a finite number of parameters. Its goal is to estimate those parameters. Examples include the T-test and ANOVA.

Non-Parametric Inference Basics

The distribution of the population is not known. It is used to test the assumptions of parametric methods, for example, to check if the population distribution is normal.

What is a Statistic?

A random variable function of the sample that does not depend on the unknown parameter.

Understanding Estimators

A statistic whose values are acceptable for estimating an unknown parameter.

Unbiasedness in Estimation

We do not allow systematic overestimation or underestimation of the parameter, which would result... Continue reading "Statistical Inference & Hypothesis Testing Concepts" »

De Morgan's Law

De Morgan's Law: (Flip if the union is true)

, image of set: [min, max]; one-to-one: horizontal line test; Onto: Image must equal domain; Bijective: one-to-one and Onto

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Possible Outcomes and Probability Calculations

• Repetition formula: n^k
  • Example: 5 awards (k) and 30 students (n), with no limit to awards per student.
• Permutation formula: P(n, k) = n! / (n - k)!
  • Example: Each student gets 1 award, so the number of students decreases by one each award.
• No overlap probability: P(n, k) / repetition formula
• Arrangements: a = slots → a! can be multiplied by arrangements within slots
• Die sum probability:
  • List combinations that lead to the sum for each die.
  • If a die is rolled multiple times, each combination has (rolls)! permutations.
  • Add

Probability Distributions for Discrete Events

The following table matches common scenarios to their appropriate probability distributions: